Abstract
This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude ε>0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes Xε turns out to be exponentially ergodic with equilibrium distribution με in total variation distance which extends a result from [60] to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index α>3 2. That is to say, in this setting we identify a deterministic time scale tεcut satisfying tεcut→∞, as ε→0, and a respective time window, tεcut±o(tεcut), during which the total variation distance between the current state and its equilibrium με essentially collapses as ε tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in [9] to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for α>3 2. The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.
Highlights
Speaking the term cutoff phenomenon with respect to a distance d1 refers to the following asymptotic dynamics: consider the setting of a parametrized family of stochastic processes (Xε)ε>0, Xε = (Xtε)t 0, such that for each ε > 0 the process Xε has a unique limiting distribution με
The proof is given in Subsection D. (2) In Definition 1.3 we introduce the class of locally layered stable process, which are precisely the class of processes for which the shortrange behavior in Theorem 3.1 in [42] remains valid
Appendix D yields the proof of Theorem 1, which implies the exponential ergodicity of Xε towards με, which extends a result by [60] to the case of an arbitrary positive finite moment
Summary
If – in addition – the limiting distributions turns out to be rotationally invariant, the existence of a cutoff profile is shown to be equivalent to a computational linear algebra eigenvector problem first established in [6] for the easier situation of the Wasserstein distance. This characterization is given as a specific orthogonality condition of the (generalized) eigenvectors of the linearization −Db(0) of −b in the stable state 0. For a complete comparison of the different settings and results and in order to avoid a lengthy introduction, we refer to the following self-explanatory Table 1.1
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
